Directory UMM :Journals:Journal_of_mathematics:EJQTDE:

Electronic Journal of Qualitative Theory of Differential Equations
2010, No. 73, 1-9; http://www.math.u-szeged.hu/ejqtde/

Note on an anisotropic
p -Laplacian equation in Rn∗
Said El Manouni
Department of Mathematics, Faculty of Sciences
Al-Imam University
P.O. Box 90950, Riyadh 11623, Saudi Arabia
Email: samanouni@imamu.edu.sa & manouni@hotmail.com
Abstract
In this paper, we study a kind of anisotropic p-Laplacian equations
in Rn . Nontrivial solutions are obtained using mountain pass theorem
given by Ambrosetti-Rabinowitz [1].
Key Words and Phrases: anisotropic p-Laplacian equations, nontrivial solutions.
MSC2000: Primary 35J20, Secondary 35J60, 35J70

1

Introduction

Consider the following anisotropic p-Laplacian problem in Rn
!


n
X
 ∂u pi −2 ∂u
∂

ai 
= f (x, u) in Rn
−

∂x
∂x
∂x
i
i
i
i=1

(1.1)

Each pi ∈ (1, ∞), ai = ai (x) are measurable real functions satisfying 0 <
a0 < ai (x) ∈ L∞ (Rn ) and we assume that the nonlinear function f satisfies
the subcritical growth conditions
|f (x, u)| ≤ g(x)|u|r
∗

∀(x, u) ∈ Rn × R

(1.2)

This work is supported by Al-Imam University project No. 281206, Riyadh, KSA

EJQTDE, 2010 No. 73, p. 1

n

1X 1

1
and
for some r ∈ (p+ − 1, p¯ − 1). Herein, p+ = max(p1 , . . . , pn ), =
p¯ n i=1 pi
n¯
p
p¯∗ =
with p¯ < n and g : Rn → R is a nonnegative function satisfying
n − p¯
∗

g ∈ Lω (Rn ),

ω=

p¯∗
.
p¯∗ − (r + 1)
1

(1.3)

n

For (1.1) we assume that F ∈ C (R × R), where F (x, s) =

s

f (x, t) dt and

0

there exists θ > p+ such that
0 < θF (x, s) ≤ sf (x, s),

Z

∀x ∈ Rn , ∀s ∈ R \ {0}.

(1.4)

In the isotropic case, we can refer the reader to the works by [3] and [7] where
existence and regularity results are obtained. As to anisotropic equations
with different orders of derivations in different directions involving critical
exponents with unbounded nonlinearities, to our knowledge they were not
intensively studied before, in passing, we mention the work [2]. Let us mention also that in [4] the authors have studied another class of anisotropic elliptic equations. Via an adaptation of the concentration-compactness lemma
of P.-L. Lions to anisotropic operators, they have obtained the existence of
multiple nonnegative solutions. Let us point out that in the case of bounded
domains, more work in this direction can be found in [5] where the authors
proved existence and nonexistence results for some anisotropic quasilinear
elliptic equations.
The purpose of this paper is to obtain nontrivial weak solutions using
mountain pass theorem (see e.g. [1]).
Our main result is the following.
ωp1

Theorem 1.1. Assume (1.2), (1.3), and (1.4). Moreover, if g ∈ L p1 −1 (Rn )
for some r + 1 < p1 < p¯∗ , then problem (1.1) has at least one nontrivial
solution.

2

Preliminaries

We let 1 ≤ p1 , . . . , pn < ∞ be N real
Denote by p the harmonic
Pnnumbers.
1
1
1
mean of these numbers, i.e., p = n i=1 pi , and set p+ = max(p1 , . . . , pn ),

EJQTDE, 2010 No. 73, p. 2

p− = min(p1 , . . . , pn ). We always have p− ≤ p ≤ p+ . The Sobolev conjugate
np
of p is denoted by p⋆ , i.e., p⋆ = n−p
.
Anisotropic Sobolev spaces were introduced and studied by Nikol’ski˘ı [8],
Slobodecki˘ı [9], Troisi [10], and later by Trudinger [11] in the framework of

Orlicz spaces.
Let (W, k · k) be the anisotropic Sobolev space defined by


1 ∂u
W = u ∈ W 1,¯p (Rn ) : ai pi
∈ Lpi (Rn ), i = 1, . . . , n ,
∂xi

with the dual (W ∗ , k · k∗ ) and the duality pairing h·, ·i. W is a real reflexive
Banach under the norm



 p1
n 
n Z
X
X
 1 ∂u 

 ∂u pi
i
ai pi
 dx

=
kuk =
ai 


 p n .
∂x
∂x
n
i
i
R
L i (R )
i=1
i=1

Let us recall that a weak solution of the equation (1.1) is any u ∈ W such
that


Z
n Z
X
 ∂u pi−2 ∂u ∂v

f (x, u)v dx,
(2.1)
dx =
ai 

∂x
∂x
∂x
n
n

i
i
i
R
R
i=1

for all v ∈ W .
They coincide with the critical points of the C 1 -energy functional corresponding to problem (1.1)


Z
Z
n
X
 ∂u pi
1
 dx −
Φ(u) =
ai (x) 

F (x, u) dx,
(2.2)

p
∂x
n
i
i
R
Rn
i=1

for all u ∈ W.

Remark 2.1. Let us remark that (1.3) implies ω1 + p¯r∗ + p¯1∗ = 1 which guarantee that the integral given in right side of (2.1) is well defined.
To deal with the functional framework we apply the following mountain
pass theorem [1].
Theorem 2.2. Let I be a C 1 -differentiable functional on a Banach space E
and satisfying the Palais-Smale condition (PS), suppose that there exists a
neighbourhood U of 0 in E and a positive constant α satisfying the following
conditions:
EJQTDE, 2010 No. 73, p. 3

(I1) I(0) = 0.
(I2) I(u) ≥ α on the boundary of U.
(I3) There exists an e ∈ W \U such that I(e) < α.
Then
c = inf sup I(γ(y))
γ∈Γ y∈[0,1]

is a critical value of I with Γ = {h ∈ C([0, 1]); h(0) = 0, h(1) = e}.
Let us recall that the functional I : E → R of class C 1 satisfies the PalaisSmale compactness condition (PS) if every sequence (uk )∞
k=1 ⊂ E for which
′
there exists M > 0 such that: I(uk ) ≤ M and I (uk ) → 0 strongly in I ⋆ as n
goes to infinity (called a (PS) sequence), has a convergent
subsequence.
R
Let us define the functional J : W → R by J(u) = Rn F (x, u) dx for each
u ∈ W. It follows from (1.2),
|F (x, u)| ≤

1
g(x)|u|r+1
r+1

(2.3)

for all (x, u) ∈ Rn × R. Then by standard argument we have F is in C 1 (Rn ×
R), hence we see that J is well defined and continuously Gˆateaux differentiable with
Z
′
J (u)v =
f (x, u)v dx
Rn

for all u, v ∈ W.

Proposition 2.3. J ′ is a compact map from W to W ∗ .
Proof. Let uk be a sequence in W which converges weakly to u. On one
hand, in view of H¨older’s inequality and Sobolev embedding, we obtain for
all 0 ≤ R ≤ +∞,
Z
 ω1
Z
∗
ω
(C1 kuk)r kvkp¯ ,
|g| dx
f (x, u)v dx ≤
|x|≥R

|x|≥R

ω

n

for all u, v ∈ W. Since g ∈ L (R ), we have lim

R→+∞

Z

|g|ω dx = 0. This

|x|≥R

implies with the fact that uk is a bounded sequence, for any ε, there exists
Rε > 0 such that
Z
Z
f (x, u)v dx ≤ ε
(2.4)
f (x, uk )v dx ≤ ε and
|x|≥Rε

|x|≥Rε

EJQTDE, 2010 No. 73, p. 4

holds for all k.
ωp1
On the other hand, since g ∈ L p1 −1 (Rn ) with r + 1 < p1 < p¯∗ , applying
Young’s inequality, we get
p1

p1

rp1

f p1 −1 (x, t) ≤ m(x) p1 −1 t p1 −1
p1
p
¯∗
¯∗
1 p
p¯∗ − (r + 1)
r + 1 prp−1
( ¯∗ −(r+1)
)
p1 −1 p
r+1
1
m(x)
t
≤
+
,
p¯∗
p¯∗
for all t ∈ R and a.e. x ∈ Bε = {x ∈ Rn ; |x| < Rε }. A simple calculation
p¯∗
1
shows that prp
< p¯∗ . Using the compact imbedding W 1,¯p (Bε ) into Lq (Bε )
1 −1 r+1
for all q ∈ [1, p¯∗ ) and the continuity of the Nemytskii’s operator Nf associated
p1

rp1

p
¯∗

with f p1 −1 from L p1 −1 r+1 (Bε ) in L1 (Bε ), we conclude that
Z
Z
p1
p1
p1 −1
f (x, uk )
f (x, u) p1 −1 dx,
dx →
|x| 0 there exists cε > 0 such that
∗

F (x, t) ≤ ε|t|p+ + cε |t|p¯

∀(x, t) ∈ Rn × R.

(3.1)

Using again (2.3), the relation (3.1) and continuous imbedding of W in
∗
Lp+ (Rn ) and Lp¯ (Rn ), we get for ||u|| small enough that


Z
Z
n
X
 ∂u pi
1
 dx −
F (x, u) dx,
ai 
Φ(u) =

p
∂x
n
i
i
R∗
R
i=1
∗
1
||u||p+ − Cε ||u||p+ − C ′ cε ||u||p¯
p+


1
p+
p¯∗ −p+
′
≥ ||u||
(
− εC) − C cε ||u||
,
p+

≥

for some constants C, C ′ > 0. Therefore, for 0 < ε < Cp1+ , there exist α and ρ
small enough positive constants such that Φ(u) ≥ α > 0 for all ||u|| = ρ.
Lemma 3.2. Suppose (1.2) and (1.4) hold. Then Φ(tu) → −∞ as t → +∞.
Proof. We have
1/p+

Φ(t

u) =

pi
Z
n
X
t p+

i=1



Z
 ∂u pi
 dx −
F (x, t1/p+ u) dx.
ai 

∂x
i
Rn
Rn

pi

Remark that F (x, tu) ≥ tθ F (x, u) for any t ≥ 1, this is due to the fact that
is increasing for all t > 0, we obtain
the function F (x,tu)
tθ
1/p+

Φ(t

u) =

pi
Z
n
X
t p+

i=1

pi



Z
 ∂u pi
θ/p
+
 dx − t
F (x, u) dx.
ai 
∂xi 
Rn
Rn

EJQTDE, 2010 No. 73, p. 6

Then
Φ(t1/p+ u) → −∞ as t → +∞
follows immediately since θ > p+ .
Lemma 3.3. Let uk be a Palais-Smale sequence of Φ. Then uk possesses a
subsequence converging strongly to some u ∈ W.
Proof. First we claim that the sequence uk is bounded. Indeed, Arguing
by contradiction and consider a subsequence still denoted by uk such that
||uk || → ∞. We have in view of (1.4)


Z
Z
n
X
 ∂uk pi
1
 dx −
F (x, uk ) dx
ai 
Φ(uk ) =

p
∂x
n
n
i
i
R
R
i=1


Z
Z
 ∂uk pi
1
1


≥
dx −
ai
f (x, uk )uk dx
p+ Rn  ∂xi 
θ Rn
!




Z
n Z
X
 ∂uk pi
 ∂uk pi
1
1
′
 dx +
 dx
=
ai 
hΦ (uk ); uk i −
ai 

p+ Rn  ∂xi 
θ
∂x
n
i
R
i=1


 n Z

 ∂uk pi
1 X
1
 dx + 1 hΦ′ (uk ); uk i.
−
ai 
≥
p+ θ i=1 Rn
∂xi 
θ

(3.2)

Passing to limit in (3.2) as k → ∞, we obtain
M≥



1
1
−
p+ θ

X
n Z
i=1





 ∂uk pi
1
1


dx ≥
||uk ||p− ,
ai 
−

∂xi
p+ θ
Rn

where M is the constant of Palais-Smale sequence. This gives a contradiction
since p+ < θ. Hence the sequence uk has a subsequence still denoted by uk
which converges weakly to some u ∈ W. For any pair integer (n, m) we have
!





 ∂un pi −2 ∂un  ∂um pi −2 ∂um
∂un ∂um




ai (x) 
−
−
∂xi 
∂xi  ∂xi 
∂xi
∂xi
∂xi
Rn
Z
= hΦ′ (un ) − Φ′ (um ); (un − um )i +
(f (x, un ) − f (x, um ))(un − um ) dx.

n Z
X
i=1

Rn

(3.3)

EJQTDE, 2010 No. 73, p. 7

By Palais-Smale condition and Proposition 2.3, it is easy to see that the right
side of (3.3) approaches zero. Finally, using the following algebraic relation
|ξ1 − ξ2 |r ≤ ((|ξ1|r−2 ξ1 − |ξ2 |r−2ξ2 )(ξ1 − ξ2 ))ρ/2 (|ξ1 |r + |ξ2 |r )1−ρ/2 ,
with ρ = r if 1 < r ≤ 2 and ρ = 2 if 2 < r, the monotonicity of the anisotropic
operator of problem (1.1) now gives the result. This concludes the proof of
Lemma 3.3.
Therefore lemmas 3.1, 3.2 and 3.3 fit into conditions setting of Theorem
2.2 of section 2, this guarantees the existence of at least a nontrivial weak
solution for (1.1).
Remark 3.4.
1- One can prove that each solution u of problem (1.1) satisfies u ∈ Lσ (Rn )
with p¯∗ ≤ σ ≤ ∞. This regularity result is based on an iterative procedure
given in the works [3] where similar results are obtained for the case of degenerate isotropic p-Laplacian problems.
1
2- Let us also mention that since ω + ε < pωp
, with 0 < ε < 1 small enough,
1 −1
ω+ε
the restrictive integrability condition g ∈ L (Rn ), suffices to proceed with
the iterating method and to bound the maximal norm of the solution.

References
[1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point
theory and applications, Funct. Anal. 14 (1973), 349–381.
[2] M. Bendahmane and S. El Manouni, Anisotropic Elliptic Equations in RN :
Existence and Regularity Results, Top. Meth. Nonl. Anal. 36(1) (2010), 129–
153.
[3] P. Drabek, Nonlinear Eigenvalues for p-Laplacian in RN , Math. Nachr. 173
(1995), 131–139.
[4] A. El Hamidi and J. M. Rakotoson. Extremal functions for the anisotropic
Sobolev inequalities. Ann. I. H. Poincar´e-AN 24 (2007), 741–756.
[5] I. Fragal`
a, F. Gazzola and B. Kawohl. Existence and nonexistence results for
anisotropic quasi- linear equations. Ann. I. H. Poincar´e-AN 21 (2004), 715–734.

EJQTDE, 2010 No. 73, p. 8

[6] S. El Manouni, The study of nonlinear problems for p-Laplacian in Rn via a
Ricceri’s principle, submitted.
[7] Lao Sen Yu, Nonlinear p-Laplacian problems on unbounded domains, Proceeding of A. M. S. 115, 4 (1992), 1037–1045.
[8] S. M. Nikol’ski˘ı, An imbedding theorem for functions with partial derivatives
considered in different metrics. Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958),
321–336. English translation: Amer. Math. Soc. Transl. 90 (1970), 27–44.
[9] L. N. Slobodecki˘ı, Generalized Sobolev spaces and their application to boundary problems for partial differential equations. Leningrad. Gos. Ped. Inst. Uˇcen.
Zap. 197 (1958), 54–112. English translation: Amer. Math. Soc. Transl. (2) 57
(1966), 207–275.
[10] M. Troisi, Teoremi di inclusione per spazi di sobolev non isotropi. Ricerche.
Mat., 18 (1969), 3–24.
[11] N. S. Trudinger, An imbedding theorem for H0 (G, Ω) spaces. Studia Math.,
50 (1974), 17–30.

(Received October 12, 2010)

EJQTDE, 2010 No. 73, p. 9